Asymptotic Implied Volatility at the Second Order with Application to the SABR Model

We provide a general method to compute a Taylor expansion in time of implied volatility for stochastic volatility models, using a heat kernel expansion. Beyond the order 0 implied volatility which is already known, we compute the first order correction exactly at all strikes from the scalar coefficient of the heat kernel expansion. Furthermore, the first correction in the heat kernel expansion gives the second order correction for implied volatility, which we also give exactly at all strikes. As an application, we compute this asymptotic expansion at order 2 for the SABR model.Comment: 27 pages; v2: typos fixed and a few notation changes; v3: published version, typos fixed and comments added. in Large Deviations and Asymptotic Methods in Finance, Springer (2015) 37-6

Simulation of chemical kinetics transport in the stratosphere

From symposium on stiff differentials system; Wildbad, F. R. Germany (4 Oct 1973). The residence time of any material brought into the stratosphere is quite large (order of a year or more). With the advent of supersonic transports, the environmental impact of the engine exhaust has become a major concern. A principal concern has been the possible catalytic destruction of ozone (O/sub 3/) by nitric oxide (NO). Given a fleet of SST's the injected NO concentration could exceed many times the ambient. Furthermore, the enhanced NO concentrations will greatly increase the destruction of O/sub 3/ through the socalled NO/sub x/ catalytic cycle. It has been estimated that based on certain estimates of NO production rate from SST there could be a world wide reduction in ozone of from 3% to 50%. Atmospheric nuclear tests in the 1950's and early 1960's injected substantial amounts of NO/sub x/ into the stratosphere. It would follow that an examination of the ozone measurements of that period could possibly be used to estimate the significance of NO/sub x/ catalytic destruction of ozone in the stratosphere. In order to carry out such a study one must use a time dependert model and compute the time history of perturbations from 1956 to 1970 starting with the ambient atmosphere as determined by the model. By parameterizing the processes of production and injection of NO into the stratosphere during each atmospheric nuclear test, a highly discortinuous source function was constructed for use in prediction of changes in the yearly averaged ozone measurements. (CH

Numerical Modeling of the Tangential Stress Effects on Convective Fluid Flows in an Open Cavity

Mathematical and numerical modeling of convection of fluids in a domain with a free boundary being subjected to an effect of a gas phase is widely investigated nowadays. When a gas medium generates on the free boundary the tangential stresses effected by evaporation, the additional characteristics of the convective flows should be studied. The new experiments in the frame of the scientific project CIMEX of the European Space Agency demand a mathematical modeling and preliminary investigation of the different features of the fluid flows. A stationary problem of convection in a rectangular cavity with a non-deformed thermocapillary boundary under conditions of gravity and zero-gravity is considered. The tangential forces created by the external gas flows on the free boundary are taken into account. Different kinds of a dependence of the tangential stresses on a longitudinal coordinate is investigated numerically. The simulations are carried out for a liquid filled cavities with different aspect ratio. The paper presents topology of the convective fluid flows in the conditions corresponding qualitatively to the CIMEX experiment.En ligne: http://www.springerlink.com/content/g4458n7445824218/info:eu-repo/semantics/publishe